Paraunitary filter banks form an interesting subset of perfect
reconstruction (PR) filter banks. We saw above that we get a PR filter
bank whenever the
synthesis polyphase matrix
times the
analysis polyphase matrix
is the identity matrix
, i.e., when

(12.70)

In particular, if
is the paraconjugate of
, we
say the filter bank is paraunitary.

Paraconjugation is the generalization of the complex conjugate
transpose operation from the unit circle to the entire
plane. A
paraunitary filter bank is therefore a generalization of an
orthogonal filter bank. Recall that an orthogonal filter bank
is one in which
is an orthogonal (or unitary) matrix, to
within a constant scale factor, and
is its transpose (or
Hermitian transpose).

To motivate the idea of paraunitary filters, let's first review some
properties of lossless filters, progressing from the simplest cases up
to paraunitary filter banks:

A linear, time-invariant filter
is said to be
lossless (or
allpass) if it preserves signal
energy. That is, if the input signal is
, and the output
signal is
, then we have

(12.71)

In terms of the
signal norm
(§4.10.1), this can be expressed more succinctly as

(12.72)

Notice that only stable filters can be lossless since, otherwise,
. We further assume all filters are causal for
simplicity.

It is straightforward to show that losslessness implies

(12.73)

That is, the frequency response must have magnitude 1 everywhere on
the unit circle in the
plane. Another way to express this is to
write

(12.74)

and this form generalizes to
over the entire the
plane.

The paraconjugate of a transfer function may be defined as the
analytic continuation of the complex conjugate from the unit circle to
the whole
plane:

(12.75)

where
denotes complex conjugation of the
coefficients only of and not the powers of .
For example, if
, then
. We can
write, for example,

(12.76)

in which the conjugation of
serves to cancel the outer
conjugation.

We refrain from conjugating
in the definition of the paraconjugate
because
is not analytic in the complex-variables sense.
Instead, we invert
, which is analytic, and which
reduces to complex conjugation on the unit circle.

for all
, where
denotes the
identity
matrix, and
denotes the Hermitian transpose
(complex-conjugate transpose) of
:

(12.80)

Note that
is a matrix
product of a
times a
matrix. If
, then
the rank must be deficient. Therefore, we must have
.
(There must be at least as many outputs as there are inputs, but it's
ok to have extra outputs.)

A lossless
transfer function matrix
is paraunitary,
i.e.,

(12.81)

Thus, every paraunitary matrix transfer function is unitary on
the unit circle for all
. Away from the unit circle,
paraunitary
is the unique analytic continuation of unitary
.

where
,
, and
. The polynomial
can be obtained by reversing the order of the coefficients in
,
conjugating them, and multiplying by
. (The factor
above
serves to restore negative powers of
and hence causality.) Such
filters are generally called allpass filters.

The synthesis filter bank is simply the paraconjugate of the
analysis filter bank:

(12.89)

That is, since the paraconjugate is the inverse of a paraunitary filter matrix,
it is exactly what we need for perfect reconstruction.

The channel filters
are power complementary:

(12.90)

This follows immediately from looking at the paraunitary property on the
unit circle.

When
is FIR, the corresponding synthesis filter matrix
is also FIR.

When
is FIR, each synthesis filter,
, is simply the
of its corresponding
analysis filter
:

(12.91)

where
is the filter length. (When the filter coefficients are
complex,
includes a complex conjugation as well.) This
follows from the fact that paraconjugating an FIR filter amounts to
simply flipping (and conjugating) its coefficients. As we observed in
(11.83) above (§11.5.2), only trivial FIR filters of
the form
can be paraunitary in the
single-input, single-output (SISO) case. In the MIMO case, on the
other hand, paraunitary systems can be composed of FIR filters of any
order.

The polyphase matrix
for any FIR paraunitary perfect
reconstruction filter bank can be written as the product of a
paraunitary and a unimodular matrix, where a
unimodular polynomial matrix
is any square
polynomial matrix having a constant nonzero
determinant. For example,

Thus, the Haar filter bank is paraunitary. This is true for any
power-complementary filter bank, since when
is
,
power-complementary and paraunitary are the same property. For more
about paraunitary filter banks, see Chapter 6 of
[287].